EMS 164 Solutions for Homework 4
Problems 2.3 and 2.7 in textbook
1.Dendritic segregation, or coring, is very common in alloy castings. Segregation can be removed by a
homogenizing anneal. The dendritic arm spacing is assumed to equal 2l (sinusiodal wavelength). In a
typical cored specimen, the dendritic arm spacing may be of the order of 10 ^
–
2 cm. How long should
the anneal time be at 1400 K in order to reduce the sinusoidal solute distribution, b, by about two
orders of magnitude? Take the diffusivity of the solute in the base metal at 1400 K as D= 10 ^
–
10
cm ^2 /s.
( 10 points)
β=β
0
*exp (
t/τ)
( 2 points for the equation)
Where β is the amplitude of concentration profile after time t,
β
0
is amplitude of the initial
concentration, τ is relaxation time and given by τ=l
2
/(π
2
*D
B
)
Given in the question:
2l= 10
2
cm
L= 0.01 /2= 0.005 cm
β/β
0
= 10
2
( 1 point)
D= 10
–
10
cm
2
/s.
τ=l
2
/ (
π
2
*D
B
) = 0.005
2
cm
2
/ (3.14
2
* 10
–
10
cm
2
/s )= 2.5 * 10
4
s
( 2 points for equation, 2 points for the
value)
β/β
0
= 10
2
= exp(t/ 2.5 * 10
4
s)=
t= 11.51 * 10
4
s= 115129.25 s= 31.98 hours
(3 points for the correct answer)
No points taken off if a value for question is taken wrong or 2l value is used instead of l.
2. The usual first step in adding dopant atoms to silicon to form an integrated circuit is to predeposit
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 Spring '10
 J.Groza
 Orders of magnitude, Atomic diffusion

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